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On the exponential behaviour of eigenfunctions and the essential spectrum of differential operators

Published online by Cambridge University Press:  14 November 2011

F. V. Atkinson  and
W. D. Evans
F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada
W. D. Evans
Affiliation:
Department of Pure Mathematics, University College, Cardiff, Wales, U.K.
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Synopsis

The paper deals with the differential equation

on [ 0, ∞) Where λ>0 and the coefficients qm are complex-valued with qn continuous and non-zero, w is positive and continuous and qm for m = 0, 1,…, n − 1. In the first part of the paper the exponential behaviour of any solution of (*) is given in terms of a function ρ(λ) which is roughly the distance of λ from the essential spectrum of a closed, densely denned linear operator T generated by T+ in L2(0, ∞ w). Next, estimates are obtained for the solutions in terms of the coefficients in (*). When the latter results are compared with the estimates established previously in terms of ρ(λ), bounds for ρ(λ) are obtained. From the general result there are two kinds of consequences. In the first, criteria for ρ(λ) = 0 for all All λ > 0 are obtained; this means that [0, ∞) lies in the essential spectrum of T in appropriate circumstances. The second type of consequence concerns bounds of the form ρ(λ) = O(λr) for λ → ∞ and r<1.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 3-4 , 1982 , pp. 271 - 300
Copyright
Copyright © Royal Society of Edinburgh 1982

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